Solving Equations

SymPy has equation solvers that can handle linear and non-linear algebraic equations as well as linear and non-linear multivariate equations. Depending upon the input equation, the answer returned can be symbolic or numeric. The most commonly used one is solve(). It uses the following syntax:

solve(f(x), x)

The output type of solve() varies with input, sometimes it returns a Python list and sometimes a Python dictionary. Usually, for single variable equations, it returns a list and for multivariable equations, it returns a dictionary.

Let’s explore different cases in the examples below:

Solving for a single variable#

Single solution#

Let’s solve the following equation that has a single solution:

$$2x+3=0$$

The solve() function returns a list with the only solution.

Multiple solutions#

Let’s solve the following equation, which has multiple solutions:

$$x^2-x+6=0$$

The solve() function returns a list with all the solutions. Let’s solve an equation with complex roots:

$$x^2+4x+5=0$$

The solve() function returns a list with all the solutions, which are complex numbers.

Interval solution#

Using SymPy, we can also find a solution for an interval. All we have to do is provide the interval alongside the equation

solve(f(x) > a, x)

where > a is the interval. Let’s see an example of interval solution:

In the code above (lines 10 and 11), we have computed the values of $x$ in the interval $f(x)>0$ and $g(x)>0$ respectively.

Trigonometric equations#

Using the SymPy module, we can solve trigonometric equations as well. Let’s solve the following equation in SymPy:

$$tan^2(x)-3=0$$

Solutions are in the range $-\pi \leq x < \pi$.

Solving systems of equations#

SymPy solve() is not just limited to solving a single variable equation, it can also solve a system of multivariable equations. This system of equations is to be in a list as the first argument of solve() and the variables will be in a list as the second argument of solve():

solve([eq1, eq2,...eqN], [x1, x2, x3,...xN])

Let’s solve the following system of equations:

$$x+y-z-4=0$$

$$x-2y+3z+6=0$$

$$2x+3y+z-7=0$$

Notice that the type of the sol is a dictionary where each variable is a key to its corresponding value.

Symbolic solution#

All the equations that we have solved can be solved with arbitrary coefficients as well. This technique is useful when optimizing systems. See the example below:

$$x+y-z-a=0$$

$$x-2y+3z+b=0$$

$$2x+3y+z-c=0$$

The solution that we get from these equations is in the form of a, b and c.

In the next lesson, we will learn about solving ordinary differential equations.